I was able to modify void Dictionary::add (std::string word) correctly to work with linked lists . Having trouble modifying the bool Dictionary::contains (std::string word) const . Not sure exactly how to modify it. Can someone help please?

I don't think it's the right way to implement the contains() method using binarySearch():
1, binary search needs your list to be sorted and I don't see any code you listed did sorting.
2, you try to apply binary search algorithm on a linked list, which sounds to me like eating rice with chopsticks - no offence to smart Asian guys! As a hint, I would imagine using b-search on an array, for example. (Question: why? I hope you can think for yourself.)
3, in terms of algorithm time complexity, binary search is O(logN), which is worse than the performance of, say, a hash table. In C++, you could use unordered map. However, I've heard that some (bad) implementation of unordered map actually gives you O(logN).

Last but not least, if you're somehow required to implement your dictionary using linkedlist(suggesting your tutor/boss is an idiot or he just wants you to look like an idiot), then you probably don't need to worry about time complexity at all. In that case, why not giving him/her the simplest implementation?(iterate through the list and check using a loop)